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The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic value?

By Fatou's theorem, the radial limit function $$\phi^{\ast}(\zeta):=\lim_{r\rightarrow1^{-}}\phi(r\zeta),$$ for a bounded analytic function $\phi$ on $\mathbb{D}$, exists for $m$-almost every $\zeta\in\partial\mathbb{D}$(here $m$ is a normalized Lebesgue measure on $\partial\mathbb{D}$). If $|\phi^{\ast}(\zeta)|=1$ for almost every $\zeta$, then $\phi$ is called an inner function.

An inner function can be factored as $$\phi(z)=e^{i\gamma}z^pB(z)\exp(-\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\mu(\zeta)).$$ Here $\mu$ is a positive finite measure on $\partial\mathbb{D}$ with $\mu\bot m$ and $B(z)$ is a Blaschke Product.

The Frostman shift $$\phi_a(z):=\tau_a\circ\phi(z)=\frac{\phi(z)-a}{1-\overline{a}\phi(z)}, |a|<1$$ are certainly inner functions when $\phi(z)$ is an inner function.

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It is best not to use mathmode ("dollar signs") if you merely want italic text - the spacing is not quite right, for a start. Please take this as a friendly suggestion for future use of LaTeX –  Yemon Choi Aug 24 '13 at 3:54
    
Thank you for your kindly suggestion. –  Jame Ake Aug 24 '13 at 11:28

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up vote 2 down vote accepted

If $a$ is an asymptotic value, the inner function $\phi_a$ may be a Blaschke product or not. It is not clear what do you mean by "how to chracterize it". In what terms? It is an inner function for which zero is an asymptotic value. (This characterizes it. Is this what you want?).

In a special case when all zeros of $\phi_a$ are real there is a complete answer (on the question, whether $\phi_a$ is a Blaschke product or not) in terms of critical values of $\phi_a$ (or $\phi$, does not matter). This answer is highly non-trivial, and its generalization to non-real zeros is not known. The characterization I am talking about is contained in the paper A. A. Goldberg, Some asymptotic properties of meromorphic functions, Complex variables, 37 (1998) 225--241. English translation from the Russian original published in Uchenye Zapiski Lvovskogo Gosudarstvennogo Universiteta, 38, 7, 1956. 54-74. If your library does not have these journals, I can send you a reprint of the translation. If you need it, send an e-mail to me. (Complex Variables journal is not freely available on the web).

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Thank you for your detailed comments. The generalized situation is what I want to know. Further more, if I can figure out $\phi_a(z)$ is not a Blaschke product, can I say something about the singular measure $\mu$ of its singular part?(e.g. $\mu$ is discrete or continous) –  Jame Ake Aug 24 '13 at 10:45
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Singular measure cannot be "continuous". In Goldberg's theorem, if it is present, it is a point measure. In general, it can be discrete or not. –  Alexandre Eremenko Aug 24 '13 at 15:11
    
Here "continuous" means the measure $\mu$ has no point masses, and "discrete" means it consists entirely of point masses. –  Jame Ake Aug 24 '13 at 15:25

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